Efficient Generation of Multi-partite Entanglement between Non-local Superconducting Qubits using Classical Feedback

TL;DR

The paper demonstrates constant-depth generation of non-local multipartite entanglement on an eight-qubit superconducting processor using teleportation-based protocols and fast classical feedback with latency around $150~\mathrm{ns}$. It showcases GHZ-state preparation, teleportation-based CNOT, unbounded fan-out, and entanglement swapping, providing quantitative fidelities and highlighting measurement-induced dephasing as a key limitation. By combining mid-circuit measurements with real-time conditional operations, the work illustrates both the practical benefits and current constraints of adaptive circuits for scalable quantum information processing. The findings guide future improvements in readout fidelity, spectral engineering, and decoherence mitigation to realize the full potential of teleportation-based protocols in near- and medium-term quantum devices.

Abstract

Quantum entanglement is one of the primary features which distinguishes quantum computers from classical computers. In gate-based quantum computing, the creation of entangled states or the distribution of entanglement across a quantum processor often requires circuit depths which grow with the number of entangled qubits. However, in teleportation-based quantum computing, one can deterministically generate entangled states with a circuit depth that is constant in the number of qubits, provided that one has access to an entangled resource state, the ability to perform mid-circuit measurements, and can rapidly transmit classical information. In this work, aided by fast classical field programmable gate array-based control hardware with a feedback latency of only 150 ns, we explore the utility of teleportation-based protocols for generating non-local, multi-partite entanglement between superconducting qubits. First, we demonstrate well-known protocols for generating Greenberger-Horne-Zeilinger (GHZ) states and non-local CNOT gates in constant depth. Next, we utilize both protocols for implementing a quantum fan-out gate in constant depth among three non-local qubits (i.e., controlled-NOT-NOT). Finally, we demonstrate deterministic state teleportation and entanglement swapping between qubits on opposite sides of our quantum processor. Throughout this work, we find that the fidelity of our teleportation-based protocols is limited by measurement-induced dephasing on idling spectator qubits. Therefore, our work serves as a useful study of the current benefits and limitations of teleportation-based protocols on contemporary superconducting quantum processors.

Figures (11)

• Figure 1: GHZ State Preparation in Constant Depth.(a) Adaptive circuit for preparing a four-qubit GHZ state on non-local qubits. Data qubits (black) are prepared in a superposition state with a Hadamard (H) gate and entangled with ancillae qubits (red) in a pair-wise fashion. Mid-circuit measurements of the ancillae qubits can be used to determine the parity of the data qubits, which can be decoded to determine which data qubits should be flipped via conditional $X$ gates ($X_c$) to prepare all four in a GHZ state. Measurements of the GHZ state in the computational basis for (b) two qubits, (c) three qubits, and (d) four qubits. Parity oscillations of the GHZ state for (e) two qubits, (f) three qubits, and (g) four qubits. From these results we calculate a GHZ state prep fidelity of $F_{\ket{\text{GHZ}_2}} = 0.92(1)$, $F_{\ket{\text{GHZ}_3}} = 0.67(2)$, and $F_{\ket{\text{GHZ}_4}} = 0.32(3)$. Error budgets for the (h) two-, (i) three-, and (j) four-qubit GHZ state preparation experiments. Dephasing on the spectator qubits becomes the dominant source of error as the number of qubits is increased.
• Figure 2: Teleportation-based CNOT.(a) A CNOT between non-local qubits (blue) can be implemented via gate teleportation as long as the ancillae qubits (black) are prepared in a Bell state. (b) When the ancillae qubits are also non-local, they can first be prepared in a Bell state using the constant depth GHZ state preparation protocol which utilizes additional ancillae (red). (c) Truth table for a teleportation-based CNOT between qubits 1 and 4. Here, the Bell state between ancillae qubits is prepared using only unitary operators. The measured fidelity is $F_\mathrm{tt} = 0.90(1)$. (d) Truth table for a teleportation-based CNOT between qubits 0 and 4. Here, the Bell state between ancillae qubits is prepared using the procedure shown in (b). The measured fidelity is $F_\mathrm{tt} = 0.75(1)$. (e) Error budget for the CNOT between qubits 1 and 4. (f) Error budget for the CNOT between qubits 0 and 4.
• Figure 3: Unbounded Fan-Out Gate.(a) The quantum fan-out gate is equivalent to performing a control-NOT$^N$ on $N$ target qubits (black). Similar to the teleportation-based CNOT gate, this can be implemented in constant depth by means of MCM and feed-forward control, provided one can use $N$ ancillae qubits (red) prepared in a GHZ state as an entanglement resource. (b) We implement a controlled-NOT-NOT gate between three non-local qubits using the protocol shown in (a). We first use two ancillae qubits to prepare three other qubits in a GHZ state using the protocol shown in Fig. \ref{['fig:ghz']}(a). Next, the ancillae qubits are actively reset (AR) as data qubits, and the GHZ qubits are used as ancillae for the fan-out gate. We perform computational basis rotations (B) on data qubits prior to the fan-out gate, and measure the data qubits in the computational basis after the fan-out gate to perform truth table tomography. (c) Experimental results for truth table tomography performed on the controlled-NOT-NOT gate. We measure a truth table fidelity of $F_\mathrm{tt} = 0.68(2)$. (d) Error budget for the controlled-NOT-NOT gate.
• Figure 4: Entanglement Swapping and Teleportation.(a) In a unitary circuit, swapping a state $\ket{\psi}$ between two ends of a register is achieved with a cascade of SWAP gates (the first CNOT gate in each SWAP can be omitted because the ancillae qubits start in $\ket{0}$). This is equivalent to two unbounded fan-out gates with the opposite orientation. (b) In measurement-based circuits, the state $\ket{\psi}$ can be teleported from one end of a register to another using a quantum repeater protocol with simultaneous entanglement swapping, whereby a series of Bell state preparations and Bell measurements are performed in an alternating manner, and the outcome of each MCM is used to perform a conditional operation ($X_c$ or $Z_c$, if measured in the $Z$ or $X$ basis, respectively) on the final qubit. (c) A related procedure, known as entanglement swapping, can be used to deterministically prepare a Bell state between the two end qubits in a register. If the input state is $\ket{00}$, the output state is $\ket{\Phi^+}$. (d) Classical outcomes for the teleportation of $\ket{0}$, $\ket{+}$, and $\ket{1}$ from one side of our quantum processor to the other. The probabilities of successful teleportation (measured via 1 - the total variational distance to the ideal distribution) are 88.1%, 99.4%, and 94.4% respectively. (e) The Pauli transfer matrix (PTM) for the teleportation protocol in (b). The ideal matrix should be $\text{diag(1, 1, 1, 1)}$, since the input and output states should be identical. The low $X$ and $Y$ eigenvalues of the PTM ($\sim0.5$) suggest there is strong (measurement-induced) dephasing. (f) Parity oscillations for the deterministic preparation of $\ket{\Phi^+}$ and $\ket{\Phi^-}$ using the protocol in (c), giving Bell state fidelities of $F_{\ket{\Phi^+}} = 0.57(1)$ and $F_{\ket{\Phi^-}} = 0.55(1)$. The small contrast in the oscillations additionally suggests a loss of phase coherence during the MCMs.
• Figure 5: Quantum Processing Unit & Frequencies.(a) Eight-qubit superconducting transmon processor. Qubits are labeled in green, individual drive lines are labeled in blue, individual readout resonators (RO) are labeled in red, and the multiplexed readout bus (MRB) is labeled in cyan. The qubits are coupled to nearest neighbors in a ring geometry via coupling resonators (CR, purple). (b) Qutrit frequency spectrum. The solid lines and dashed lines denote the GE and EF transition of each qutrit, respectively. (b) Readout resonator frequency spectrum. The frequency of the readout resonator coupled to each qubit is shown by a dashed line.